Aspects of Statistical Estimation

Introduction

The process of estimating a population parameter, unknown and fixed, from a sample statistic, known and random, is called statistical estimation. This process involves forming either a point estimate or an interval estimate. The former is a single numeric value, often referred to as the “best guess” at estimating the value of the corresponding population parameter.

Discussion

In contrast, the latter is a range of plausible values that estimate population parameters and is given as a point estimate ± margin of error. The margin of error, E, is a value by which a point estimate, within an interval estimate, might deviate from the real population value (Loftus, 2022). To minimize the margin of error, one should choose a reasonably large sample size to reflect the population parameter accurately. The margin of error is given by

The z-score tells where a point estimate lies on a normal distribution curve; σ is the sample standard deviation, the amount of variation from the mean for a single sample; n is the sample size. The quotient of σ and n is called standard error,.

A confidence interval is a range of estimates corresponding to the true population parameter with a specified degree of confidence. Confidence limits are the values at the lower and upper ends of a q-percent confidence interval, while confidence coefficient is the number of standard deviations on either side of the sample mean within this q-percent confidence interval. In other words,

Confidence interval = Point estimate ± Confidence coefficient * Standard Error.

For example, a random sample of 250 households surveyed in a large city found that 170 of them own at least one pet. Using these concepts, one can estimate the 95% confidence interval for a population parameter, such as the percentage of households in the city who own at least one pet. The process is as follows:

1. The point estimate, p, is the sample proportion: 170/250 = 0.68.
2. The standard error: 
3. The confidence coefficient is the z-score for the 97.5^th percentile, or 1.96.
4. The margin of error: E = k*

   

    = 1.96 * 0.0295 = 0.0578, or 5.78%.
5. The interval estimate is between p ± E, or 0.68-0.0578 = 0.62 and 0.68+0.0578 = 0.74.

Conclusion

Thus, percentage-wise, the 95% confidence interval for the specified population parameter is between 62% and 74%.

Reference

Loftus, S. C. (2021). Basic Statistics with R: Reaching Decisions with Data. Academic Press.